Cooperative binding systems are being studied taking into account site or subunit interactions, ligand interactions, aggregation and redistribution in proteins and model systems. Methods are being developed to evaluate reasonable parameters describing these reactions. We study amino acids and proteins where the number of sites is small, and also those systems such as the protein-DNA complex and the helix-coil transition where the number of sites can be very large. An additional system where the number of sites is large is that involved in the subunit interactions of proteins. The interactions between the four subunits of hemoglobin can now be studied in detail. Five sets of X-ray data on the structure of human deoxy hemoglobin A now reside in the Brookhaven data banks. Two of these sets show the effects of polyethylene glycol on the structure of deoxy hemoglobin. These data have been analyzed to quantify the distances between the interacting groups as well as the water bridges in the dimer interface of deoxy hemoglobin. Our results lead us to believe that the main source of the cooperativity on the binding of oxygen to hemoglobin derives from changes in the pK values of the ionizable groups in the dimer interface. We now propose that the linked function model is the most reasonable one for explaining the cooperative phenomena on the binding of oxygen to hemoglobin.